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Properties

Label 20.0.27876940985...6176.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 439^{10}$
Root discriminant $41.90$
Ramified primes $2, 439$
Class number $375$ (GRH)
Class group $[5, 75]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12544, 0, -91552, 0, 227145, 0, -99224, 0, 15140, 0, -5112, 0, 7134, 0, -712, 0, 180, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^18 + 180*x^16 - 712*x^14 + 7134*x^12 - 5112*x^10 + 15140*x^8 - 99224*x^6 + 227145*x^4 - 91552*x^2 + 12544)
 
gp: K = bnfinit(x^20 - 8*x^18 + 180*x^16 - 712*x^14 + 7134*x^12 - 5112*x^10 + 15140*x^8 - 99224*x^6 + 227145*x^4 - 91552*x^2 + 12544, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{18} + 180 x^{16} - 712 x^{14} + 7134 x^{12} - 5112 x^{10} + 15140 x^{8} - 99224 x^{6} + 227145 x^{4} - 91552 x^{2} + 12544 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(278769409858424250725488500146176=2^{20}\cdot 439^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.90$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 439$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{2}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{6} - \frac{1}{16} a^{4} + \frac{1}{16} a^{2}$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{32} a^{4} + \frac{1}{32} a^{3} + \frac{15}{32} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{10} - \frac{1}{32} a^{6} - \frac{1}{4} a^{3} - \frac{15}{64} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{128} a^{11} - \frac{1}{64} a^{7} + \frac{1}{128} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{256} a^{12} - \frac{1}{256} a^{11} + \frac{3}{128} a^{8} + \frac{1}{128} a^{7} + \frac{1}{32} a^{6} - \frac{1}{16} a^{5} - \frac{7}{256} a^{4} - \frac{1}{256} a^{3} + \frac{15}{32} a^{2} + \frac{1}{16} a - \frac{1}{2}$, $\frac{1}{256} a^{13} - \frac{1}{256} a^{11} - \frac{1}{128} a^{9} + \frac{1}{128} a^{7} - \frac{1}{16} a^{6} - \frac{15}{256} a^{5} - \frac{1}{256} a^{3} + \frac{1}{16} a^{2} + \frac{1}{16} a$, $\frac{1}{512} a^{14} - \frac{1}{512} a^{13} - \frac{1}{512} a^{12} + \frac{1}{512} a^{11} + \frac{1}{256} a^{10} + \frac{1}{256} a^{9} + \frac{1}{256} a^{8} + \frac{7}{256} a^{7} + \frac{25}{512} a^{6} + \frac{15}{512} a^{5} + \frac{63}{512} a^{4} - \frac{15}{512} a^{3} + \frac{57}{128} a^{2} - \frac{1}{32} a + \frac{3}{8}$, $\frac{1}{512} a^{15} + \frac{1}{512} a^{11} - \frac{1}{128} a^{10} - \frac{1}{32} a^{8} + \frac{11}{512} a^{7} + \frac{3}{64} a^{6} - \frac{1}{32} a^{5} - \frac{3}{32} a^{4} - \frac{13}{512} a^{3} - \frac{5}{128} a^{2} + \frac{1}{32} a + \frac{1}{8}$, $\frac{1}{2048} a^{16} - \frac{3}{2048} a^{12} + \frac{3}{2048} a^{8} - \frac{1}{16} a^{6} + \frac{255}{2048} a^{4} - \frac{1}{4} a^{3} + \frac{5}{16} a^{2} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{14336} a^{17} - \frac{3}{3584} a^{15} - \frac{3}{14336} a^{13} - \frac{1}{512} a^{11} - \frac{1}{128} a^{10} - \frac{125}{14336} a^{9} - \frac{1}{32} a^{8} - \frac{89}{3584} a^{7} + \frac{3}{64} a^{6} - \frac{705}{14336} a^{5} - \frac{3}{32} a^{4} - \frac{59}{512} a^{3} - \frac{5}{128} a^{2} - \frac{19}{224} a + \frac{1}{8}$, $\frac{1}{79367011913728} a^{18} + \frac{6558735731}{79367011913728} a^{16} + \frac{46026157125}{79367011913728} a^{14} - \frac{1}{512} a^{13} - \frac{12166573031}{11338144559104} a^{12} - \frac{1}{512} a^{11} + \frac{7974818105}{7215182901248} a^{10} + \frac{1}{256} a^{9} - \frac{1163713941431}{79367011913728} a^{8} - \frac{7}{256} a^{7} + \frac{269477451317}{7215182901248} a^{6} - \frac{17}{512} a^{5} + \frac{140440148355}{11338144559104} a^{4} + \frac{15}{512} a^{3} + \frac{26016066645}{155013695144} a^{2} + \frac{1}{32} a + \frac{10677205789}{44289627184}$, $\frac{1}{158734023827456} a^{19} - \frac{1}{158734023827456} a^{18} + \frac{1022532333}{158734023827456} a^{17} + \frac{32194688055}{158734023827456} a^{16} + \frac{112460597901}{158734023827456} a^{15} + \frac{108987538019}{158734023827456} a^{14} - \frac{223571096167}{158734023827456} a^{13} + \frac{17702776429}{22676289118208} a^{12} - \frac{48393798311}{14430365802496} a^{11} + \frac{20209490103}{14430365802496} a^{10} + \frac{2318557995911}{158734023827456} a^{9} + \frac{3450165944805}{158734023827456} a^{8} + \frac{138621734637}{14430365802496} a^{7} + \frac{308300866947}{14430365802496} a^{6} - \frac{229347505677}{158734023827456} a^{5} - \frac{478148555633}{22676289118208} a^{4} + \frac{1665712744039}{9920876489216} a^{3} - \frac{629400897143}{4960438244608} a^{2} - \frac{99650813839}{310027390288} a + \frac{33612421395}{88579254368}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}\times C_{75}$, which has order $375$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1562135}{15651155968} a^{19} + \frac{12119549}{15651155968} a^{17} - \frac{278186235}{15651155968} a^{15} + \frac{1044720857}{15651155968} a^{13} - \frac{10880955317}{15651155968} a^{11} + \frac{5355396055}{15651155968} a^{9} - \frac{21984419785}{15651155968} a^{7} + \frac{151275534675}{15651155968} a^{5} - \frac{19705265623}{978197248} a^{3} + \frac{14696877}{3821083} a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 299443573.122 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{439}) \), \(\Q(\sqrt{-439}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{439})\), 5.5.3083536.1 x5, 10.10.16696389126347776.1, 10.0.4174097281586944.1 x5, 10.0.38032777053184.4 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
439Data not computed